Explanation Given: A model for the spread of an infectious disease among mice is given by d N d t = r N − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] where N is the size of the population of mice, α is the mortality rate due to infection, b is the mortality rate due to natural causes, β is the transmission coefficient for the rate that infected mice infect susceptible mice, v is the rate mice recover from infection and γ is the rate that mice lose immunity. Calculation: It is given that the equation is d N d t = r N − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] . Rewrite the given equation as follows: d N d t − r N = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ...... ( 1 ) Now, it is observed that the above equation is in the form of first order differential equation. Therefore, the integrating factor is calculated as follows: I = e ∫ − r d t = e − r t Multiply the integrating factor I = e − r t in equation (1) as follows: e − r t ( d N d t − r N ) = e − r t ( − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ) e − r t ⋅ d N d t − e − r t ⋅ ( − r N ) = e − r t ( − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ) D t ( N e − r t ) = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] e − r t Integrate the above equation. ∫ D t ( N e − r t ) = ∫ − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] e − r t N e − r t = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ∫ e − r t N e − r t = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] [ − 1 r e − r t ] + C N ( t ) = α ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] + C e r t The obtained equation is given below

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 10.2, Problem 26E

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To show: The solution of the equation dNdt=rN−αr(α+b+v)β[α−r(1+vb+γ)] for the spread of an infectious disease among mice is N(t)=(α+b+ν)βR[(R−α)ert+α] where R=α −r(1+νb+γ).

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Chapter 10.2, Problem 25E

Chapter 10.2, Problem 27E

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Chapter 10 Solutions

Calculus with Applications (11th Edition)

Ch. 10.1 - Find all solutions of the differential equation . Ch. 10.1 - Prob. 2YT Ch. 10.1 - Prob. 3YT Ch. 10.1 - Prob. 4YT Ch. 10.1 - Prob. 1WE Ch. 10.1 - Prob. 2WE Ch. 10.1 - Prob. 3WE Ch. 10.1 - Prob. 4WE Ch. 10.1 - Prob. 5WE Ch. 10.1 - Find the general solution for each differential...

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The solution of the equation d N d t = r N − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] for the spread of an infectious disease among mice is N ( t ) = ( α + b + ν ) β R [ ( R − α ) e r t + α ] where R = α − r ( 1 + ν b + γ ) .

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To show: The solution of the equation dNdt=rN−αr(α+b+v)β[α−r(1+vb+γ)] for the spread of an infectious disease among mice is N(t)=(α+b+ν)βR[(R−α)ert+α] where R=α −r(1+νb+γ).

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Chapter 10 Solutions